Question: In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$  Compute $\frac{[CEF]}{[DBE]}.$

[asy]
unitsize(0.025 cm);

pair A, B, C, D, E, F;

B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);

draw(C--B--A--F--D);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, NE);
label("$D$", D, W);
label("$E$", E, SW);
label("$F$", F, SE);
[/asy]
Solution: Since $AB = 115$ and $AD = 38,$ $BD = 115 - 38 = 77.$

[asy]
unitsize(0.025 cm);

pair A, B, C, D, E, F;

B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);

draw(C--B--A--F--D);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, NE);
label("$D$", D, W);
label("$E$", E, SW);
label("$F$", F, SE);
label("$38$", (A + D)/2, NW, red);
label("$77$", (B + D)/2, NW, red);
label("$115$", (A + C)/2, NE, red);
label("$77$", (C + F)/2, NE, red);
[/asy]

Let $\mathbf{a}$ denote $\overrightarrow{A},$ etc.  Since $AD = 38$ and $BD = 77,$
\[\mathbf{d} = \frac{77}{115} \mathbf{a} + \frac{38}{115} \mathbf{b}.\]Since $AC = 115$ and $CF = 77,$
\[\mathbf{c} = \frac{77}{192} \mathbf{a} + \frac{115}{192} \mathbf{f}.\]Isolating $\mathbf{a}$ in each equation, we obtain
\[\mathbf{a} = \frac{115 \mathbf{d} - 38 \mathbf{b}}{77} = \frac{192 \mathbf{c} - 115 \mathbf{f}}{77}.\]Then $115 \mathbf{d} - 38 \mathbf{b} = 192 \mathbf{c} - 115 \mathbf{f},$ so $115 \mathbf{d} + 115 \mathbf{f} = 38 \mathbf{b} + 192 \mathbf{c},$ or
\[\frac{115}{230} \mathbf{d} + \frac{115}{230} \mathbf{f} = \frac{38}{230} \mathbf{b} + \frac{192}{230} \mathbf{c}.\]Since the coefficients on both sides of the equation add up to 1, the vector on the left side lies on line $DF,$ and the vector on the right side lies on line $BC.$  Therefore, this common vector is $\mathbf{e}.$

Furthermore, $\frac{EF}{DE} = \frac{115}{115} = 1$ and $\frac{CE}{BE} = \frac{192}{38} = \frac{96}{19},$ so
\begin{align*}
\frac{[CEF]}{[DBE]} &= \frac{\frac{1}{2} \cdot EF \cdot CE \cdot \sin \angle CEF}{\frac{1}{2} \cdot DE \cdot BE \cdot \sin \angle BED} \\
&= \frac{EF}{DE} \cdot \frac{CE}{BE} \cdot \frac{\sin \angle CEF}{\sin \angle BED} \\
&= \boxed{\frac{19}{96}}.
\end{align*}